Project Summary/Abstract Functional biomarkers that are in the form of curves, images, and objects, are often collected in biomedical studies nowadays, due to the rapid advances in data acquisition technology. In recent research, there has been growing awareness that the underlying association between the functional biomarkers and outcomes may be prone to considerable heterogeneity. Those heterogeneous associations often shed insight on scienti?c discoveries and entail signi?cant implications, but tend to be overlooked by many existing functional data analysis procedures with a narrow focus on the response mean. Our proposal aims at offering researchers alternative tools to explore the comprehensive information of the relationship between functional biomarkers and survival time. In contrast to statistical models that presume constant effects of covariates, quantile regression (QR) ac- commodates varying effects and may reveal more detailed dependence structure of outcomes on covariates. Regretfully, QR for functional data has barely been studied. The objective of this proposal is to make the QR framework applicable for investigating the regression heterogeneity in functional survival data, to develop reliable and ef?cient estimation approaches, and to obtain sharp inference on the effects of functional biomarkers. We will propose a ?local? functional censored QR (FCQR) method to evaluate the impacts of functional biomarkers on the survival time at a single or multiple pre-speci?ed quantile levels and develop a related signi?cance test for testing the impact of functional biomarkers (Aim 1). Then we will develop a ?global? FCQR method to investigate the varying effects of functional biomarkers on the survival time over a region of quantile levels, which will provide researchers with a comprehensive picture about the covariates-response association. In addition, two inference procedures, including a bootstrap resampling method for estimating the standard errors and the martingale-based model diagnostics, will be developed (Aim 2). Moreover, we will extend the ?local? FCQR method to longitudi- nal measurements of functional biomarkers for dynamic prediction of residual life (Aim 3). Also, we will develop statistical software that ef?ciently implements the proposed methods. The innovation of our proposal is at least three-fold. Firstly, it will produce reliable and ef?cient FCQR tools that facilitate the identi?cation and evaluation of new valuable functional biomarkers. Secondly, the successful completion of my proposal can signi?cantly advance the theory of QR and semi/non-parametric statistics, and further, broaden their applications in lots of biomedical research. Thirdly, our proposed methods will also serve as a ?exible platform for examining the heterogeneity in functional data. They can be readily extended to other public health applications.